回覆 25# 余師傅 的帖子
計算第 4 題
易知 P_1(a，c)，P_2(a^2 + bc，ac + cd)
(1) c = 0
OP_1 = 1，a^2 + c^2 = 1，a = ±1
(i) a = 1，P_n = (1，0)，OP_n = 1
(ii) a = -1，P_n = (±1，0)，OP_n = 1
(2) c ≠ 0
det(A) = 1，ad - bc = 1
OP_1 = 1，a^2 + c^2 = 1
OP_2 = 1，(a^2 + bc)^2 + (ac + cd)^2 = 1
a^4 + 2a^2bc + b^2c^2 + a^2c^2 + 2ac^2d + c^2d^2 = 1
a^4 + 2a^2(ad - 1) + (ad - 1)^2 + a^2(1 - a^2) + 2ad(1 - a^2) + (1 - a^2)d^2 = 1
d^2 = a^2
(i) d = a，a^2 - bc = 1 = a^2 + c^2，b = -c，矩陣 A 是旋轉矩陣，OP_n = 1
(ii) d = -a，P_1(a，c)，P_2(-1，0)，P_3(-a，-c)，P_4(1，0)，P_5(a，c)，P_6(-1，0)，...，OP_n = 1
